My neighbor has an old Quadra 610 that won't boot and gives off the death chime. She uses it for word processing. The screen is blank. I have tested out the monitor, and that is not the problem.
If you have MacTracker, the sound if under the Desktops/Classic Macintoshes/Macintosh Centris 610. Then it is has Death Chime. Click it an it plays. it is rather long and contains many notes.
Is it the "arpeggio of death" (four notes that sound ironically happy)? I'm pretty sure that's a sign of a failure during the power-up self test. If I'm not mistaken, it's likely to be a hardware failure.
As for what you should do ... hard to say; it could be any number of things. Easiest thing (barring money) is to find maybe a used Mac and replace the aging machine.
Try pulling out RAM modules one at a time. The other possibility is something on the SCSI bus has died, so try booting from a CD with the HD disconnected and vice-versa. I'm not sure if the external SCSI connector has it's own bus, but you could try booting from an external drive with nothing attached to the internal bus. If none of this works, check eBay for a replacement motherboard. This Powermac 6100 motherboard for $13.50 looks like a cheap and easy way to go, and would involve a significant speed boost too.